Here's a partially-filled Hessian matrix. $\begin{bmatrix} -y\sin(x) & \cos(x) & 4 \\ \\ \cos(x) & -2z & ??? \\ \\ 4 & -2y & 0 \end{bmatrix}$ What is the missing entry? Choose 1 answer: Choose 1 answer: (Choice A) A $2z$ (Choice B) B $-2y$ (Choice C) C $\cos(x)$ (Choice D) D There's not enough information.
The Hessian of a scalar field $f$ is the matrix that contains all its second-order partial derivative information. $\bold{H}(f) = \begin{bmatrix} f_{xx} & f_{xy} & f_{xz}\\ \\ f_{yx} & f_{yy} & f_{yz} \\ \\ f_{zx} & f_{zy} & f_{zz} \end{bmatrix}$ Because the order of mixed partial derivatives often doesn't matter, the Hessian matrix is usually symmetric. We can use this fact to find $f_{yz}$, which is equal to $f_{zy}$. Matching to the bottom middle of the matrix, the missing entry is $-2y$.